Advanced Geometry
Similarity
Lesson 4
Proportional Parts
Triangle Proportionality Theorem
If a line is parallel to one side of a triangle and intersects
the other two sides in two distinct points, then it separates
these sides into segments of proportional lengths.
If BD AE ,
CB CD

.
BA DE
Midsegment
endpoints are the midpoints of two sides
Triangle Midsegment Theorem
A midsegment of a triangle is parallel to one side of the triangle,
and its length is one-half the length of that side.
BD AE
and
1
BD  AE
2
Example:
Find x, BD, and AE.
Proportional Segments
If three or more parallel lines intersect two transversals,
then they cut off the transversals proportionally.
FJ GK HL
FG JK

GH KL
Example:
Find x.
Proportional Perimeters
If two triangles are similar, then their perimeters are
proportional to the measures of the corresponding sides.
EXAMPLE:
If ∆DEF ∼ ∆GFH, find the perimeter of ∆DEF.
Special Segments of Similar Triangles
If two triangles are similar, then the measures of the
corresponding altitudes, angle bisectors, and medians
are proportional to the measures of the corresponding sides.
EXAMPLE:
In the figure, ∆EFD ~ ∆JKI. EG is a median of ∆EDF and
JL is a median of ∆JIK. Find JL if EF = 36, EG = 18, and
JK = 56.
EXAMPLE: The drawing below illustrates two poles supported by
wires. ∆ABC ~ ∆GED. AF  CF and FG  GC  DC.
Find the height of pole EC.
Angle Bisectors
An angle bisector in a triangle separates the opposite side into
segments that have the same ratio as the other two sides.
CD
AD

CB
AB
segments
with
endpoint A
segments
with
endpoint C
EXAMPLE:
Find x if AB = 10, AD = 6, DC = x, and BC = x + 6.